There is currently a tremendous amount of activity directed toward the discovery and optimization of materials and material systems such as phosphors, polymers, pharmacological compounds, semiconducting solids, and devices and the like. These new materials are typically useful because they have superior values for one or several properties. These properties could include (but are not limited to) electrical conductivity, color, bio-inertness, fabrication cost, or any other property. A variety of fields (pharmacology, chemistry, materials science) focus on the development of new materials and devices with superior properties. Unfortunately, even though the chemistry of both small molecules and extended solids has been extensively explored, few general principles have emerged that allow one to predict with certainty the composition, structure, and reaction pathways for synthesis of such materials. New materials are typically discovered through experimentation, rather than designed from existing principles.
The ability to discover new materials presupposes (1) the ability to actually make the material, and (2) the ability to accurately measure the properties of interest, or other properties that correlate with the properties of interest. Development of a material with superior properties that correlate with the properties of interest. Development of a material with superior properties also requires (3) the ability to make materials that are different in some way—meaning that the materials are in some sense not identical, whether in composition, molecular structure, processing history, raw material source, or any other difference that might impact a material's properties—and (4) a way to compare the properties of the different materials.
A common challenge is understanding how two materials actually differ from each other. Any two materials might be similar in one or many ways (e.g., composition) but different in many other ways. Thus, the properties of one material might be “better” (for a particular purpose) than those of another material for any number of reasons. One goal of experimental science is determining how properties vary with different parameters. In this sense, a parameter is any variable whose value can change in either a continuous or discontinuous fashion. Parameters can include concentrations of different chemical species (e.g., elements, compounds, solvents), temperature, annealing time, molecular weight, exposure time to radiation, process sequence or any other variable. Experimental studies typically examine the variation of a given property (e.g., smell) with a measured parameter (e.g., molecular weight), often with the implicit assumption that all other parameters are held constant (i.e., their values are identical for the compared samples). In the ideal case, two materials only differ in one parameter, and variation in the measured property is construed to be caused by variation in this parameter.
Unfortunately, it is difficult or impossible to completely determine how two materials are “different”. While variation in a given parameter (e.g., chemical composition) might be fairly obvious (e.g., one sample has 20% more nitrogen than the other), variation in another parameter might remain hidden (e.g., one sample has a slightly preferred grain orientation, vs. another sample's random orientation). The challenge is determining which parameters have a significant effect on the property of interest. This challenge requires the examination of the effects of many different parameters on the desired properties. Variation in each of these parameters creates a parameter space: a high-dimensional space defined by all the relevant parameters that describe a material. A single material is thus defined by its coordinates within this parameter space—the values for each of these parameters for the given material. The goal of materials development is finding the coordinates of the material with the best set of desired properties. The commonly used analogy “looking for a needle in a haystack” can loosely describe this process: the parameter space is the “haystack”, and the material(s) with the best set of properties is (are) the needle(s).
Traditionally, the discovery and development of various materials has predominantly been a trial and error process carried out by scientists who generate data one experiment at a time—in other words, each axis in the parameter space is examined serially. This process suffers from low success rates, long time lines, and high costs, particularly as the desired materials increase in complexity. Nevertheless, these methods have been successful for developing materials whose properties are governed by a relatively small number of parameters.
However, many properties can be a function of a large number of different parameters. Additionally, the combined effects of parameter variation can be much more complicated than the discrete effects of varying one or two parameters by themselves. For such a property, a very large parameter space must be examined in order to find the material with the best properties. As a result, the discovery of new materials often depends largely on the ability to synthesize and analyze large numbers of new materials over a very broad parameter space. For example, one commentator has noted that to search the system of organic compounds of up to thirty atoms drawn from just five elements—C, O, N, S and H—would require preparing a library of roughly 1063 samples (an amount that, at just 1 mg each, is estimated to require a total mass of approximately 1060 grams—roughly the mass of 1027 suns). See W. F. Maier, “Combinatorial Chemistry—Challenge and Chance for the Development of New Catalysts and Materials,” Angew. Chem. Int. Ed., 1999, 38, 1216. When material characteristics vary as a function of process conditions as well as composition, the search becomes correspondingly more complex. One approach to the preparation and analysis of such large numbers of compounds has been the application of combinatorial methods.
In general, combinatorics refers to the process of creating vast numbers of discrete, diverse samples by varying a set of parameters in all possible combinations. Since its introduction into the bio- and pharmaceutical industries in the late 80's, it has dramatically sped up the drug discovery process and is now becoming a standard practice in those industries. See, e.g., Chem. Eng. News, Feb. 12, 1996. Only recently have combinatorial techniques been successfully applied to the preparation of materials outside of these fields. See, e.g., E. Danielson et al., SCIENCE 279, pp. 837-839, 1998; E. Danielson et al., NATURE 389, pp. 944-948, 1997; G. Briceno et al., SCIENCE 270, pp. 273-275, 1995; X. D. Xiang et al., SCIENCE 268, 1738-1740, 1995. By using various rapid deposition techniques, array-addressing strategies, and processing conditions, it is now possible to generate hundreds to thousands of diverse materials on a substrate of only a few square inches. These materials include, e.g., high Tc superconductors, magnetoresistors, and phosphors. Using these techniques, it is now possible to create large libraries of chemically diverse compounds or materials, including biomaterials, organics, inorganics, intermetallics, metal alloys, and ceramics, using a variety of sputtering, ablation, evaporation, and liquid dispensing systems as disclosed, for example, in U.S. Pat. Nos. 5,959,297, 6,004,617, 6,030,917 and 6,045,671, and U.S. application Ser. No. 09/119,187, filed on Jul. 20, 1998, each of which is incorporated by reference herein.
An implicit goal of any experimental study is getting the most information for the minimum cost (including time); this goal is especially stringent for large parameter spaces that require vast numbers of experiments. This requires (1) maximizing the information content of each experimental point, and (2) minimizing the resource cost to synthesize and measure each experimental point. The process of deciding where in the parameter space to make and measure samples is called “sampling” or “populating” the parameter space. This process requires choosing a plurality of points in the space representing materials for synthesis and measurement. A subsequent, equally important requirement is actually making and measuring samples with the desired coordinates.
As discussed previously, the parameter spaces to which combinatorial methods are typically applied are often very large. Additionally, small changes in the values of parameters can have a large change on properties. As a result, the effective design and preparation of combinatorial libraries is a crucial factor in the success of a combinatorial project. This requirement (the process of choosing points for experimentation that have the most information at lowest cost) is described herein as efficient sampling of the parameter space. The goal of efficient sampling is choosing the minimum number of points for evaluation (synthesis and measurement) while still achieving a material with the desired set of properties. While efficient sampling is of course important for low dimensional parameter spaces, it is critical for cost effective exploration of high dimensional parameter spaces.
Regardless of the dimensionality of the relevant parameter space, historical experimentation has almost always been based upon synthesis and measurement of lower dimensional spaces (e.g., slices or projections). The ease with which humans interpret graphical data has led to the design of most experiments as evaluation of the response of a single dependent variable (y) on a single independent variable (x). Indeed, scientists using combinatorial methods have often designed combinatorial libraries by transposing a two dimensional projection from the parameter space onto a (two-dimensional) plane. For a given N-dimensional parameter space, N−2 parameters are constrained by the scientist, such that only 2 parameters vary independently across the library. This variation may be achieved by creating a set of gradients that define composition change across the library, or by defining a set of linear equations for distributing components to various locations on the substrate, or other ways.
Because the dimensionality of the projection is the same as the dimensionality of the substrate (i.e., a dimensionality of two), it is often easy to correlate the variation of points across the library with variation across the parameter space, which can aid interpretation. Additionally, it might often be relatively easier to perform the physical synthesis process (i.e., make the library) when the parameter space is sampled using projections. As a result, many combinatorial libraries are made by directly transposing different two-dimensional projections onto a two-dimensional substrate or other carrier. This method is useful for a large range of unexplored materials (e.g., ternary composition diagrams), so has found extensive use for low-dimensional parameter space explorations.
However, direct transposition of projections, whether by gradients, equations, or other methods, may not be the most efficient way to sample high-dimensional parameter spaces. Indeed, the ease with which 2-D projections can be designed, synthesized, and interpreted has often taken precedence over higher-dimensional sampling strategies that could be more efficient. Additionally, inferring the variation of properties in high-dimensional spaces using only data from multiple projections through the space can lead to erroneous conclusions for complex systems.
In summary, the sampling strategy for the vast majority of prior scientific work is a result of either human interpretive limitations (for example, not being able to “see” in high dimensions) or equipment limitations. More precisely, for many combinatorial studies, the library design process has yielded the sampling strategy, not the other way around. While this is sufficient if a given library design yields an efficient sampling, it is not optimal if the library design does not yield an efficient sampling.